A Texas healthcare worker who provided care for Thomas Eric Duncan, the first patient to be given a diagnosis of Ebola in the US, who died on Wednesday, has tested positive for the deadly virus.

At a Sunday morning press conference at the hospital, it was confirmed that a close contact of the healthcare worker – who officials said was wearing full protective gear when he or she made contact with Duncan – has also been placed, “proactively”, in isolation. The new case is the first person-to-person transmission of the disease in the US.

Dr Tom Frieden, the head of the Centers for Disease Control and Prevention (CDC), said the diagnosis of the healthcare worker showed there had been a clear breach of safety protocol at the hospital.

“We’re deeply concerned about this new development,” Frieden said on CBS. “I think the fact that we don’t know of a breach in protocol is concerning because clearly there was a breach in protocol. We have the ability to prevent the spread of Ebola by caring safely for patients.”

The worker was reported to be in stable condition in isolation at the Texas Health Presbyterian hospital in Dallas. The hospital is no longer taking any other emergency patients.

Frieden told CBS the worker had treated Duncan multiple times after the Liberian man was diagnosed, and said that all those who had treated Duncan were now considered to be potentially exposed.

At a subsequent media briefing, Frieden said the healthcare worker had not been able to identify a breach in protocol during his or her contact with Duncan. He also said the healthcare worker is not one of 48 people who have been being monitored because of contact with Duncan.

Healthcare workers treating Duncan were to follow CDC protocol that included wearing protective gear. Among the things CDC will investigate, Frieden said, is how the workers took off that gear – because removing it incorrectly can lead to a contamination.

At the hospital press conference, Dallas County judge Clay Jenkins said the healthcare worker was a “heroic person who provided care for Mr Duncan” but did not release his or her name. News of the second diagnosis broke overnight, after a preliminary blood test on the healthcare worker, who had reported a low-grade fever on Friday night.

On Sunday morning Dr Dan Varga, of Texas Health Presbyterian hospital, said the worker had been “following a self-monitoring regimen prescribed by the Centers for Disease Control and Prevention” (CDC) and that “the entire process from the patient’s self-monitoring to the admission to isolation took less than 90 minutes”.

He added: “The patient’s condition is stable. In addition, a close contact has also been proactively placed in isolation. The caregiver and the family have requested total privacy, so we can’t discuss any more details of the situation.”

Dr Varga said the hospital was now not taking any other emergency patients. Answering questions, he said the healthcare worker had been wearing full protective gear and following all CDC guidelines when he or she made contact with Duncan.

Mike Rawlings, the mayor of Dallas, addressed likely public fears brought about by the second case. He said: “We heard about this around midnight and have been working throughout the morning to make sure the citizens of Dallas are safe when they wake up. I believe I can say they are.”

Rawlings detailed protective measures taken by the city, including the Dallas fire and rescue haz-mat team “clearing up and decontaminating any of the open areas of an apartment complex” and “standing by to make sure nobody enters that apartment complex”.

“Furthermore,” he said, “we have knocked on every door in that block and talked to every person who came to the door to explain what has happened and what we have done.”

Rawlings said there was believed to be a pet inside the apartment of the healthcare worker now in isolation. He said the pet was not believed to show any signs of Ebola, and that authorities would take care of it. This week in Spain, a dog owned by an Ebola patient was euthanised.

A neighbour, Cliff Lawson, 57, told Reuters he was woken at 6am by two Dallas police officers, who told him “don’t panic”.

“I went back to bed after that. There’s nothing you can do about it. You can’t wrap your house in bubble wrap,” Lawson said.

A hospital statement detailed the steps taken by the hospital since the admission of Duncan, on 28 September, on his second visit.

“We have known that further cases of Ebola are a possibility among those who were in contact with Mr Duncan before he passed away last week,” the statement said. “The system of monitoring, quarantine and isolation was established to protect those who cared for Mr Duncan as well as the community at large by identifying any potential ebola cases as early as possible and getting those individuals into treatment immediately.”

Duncan travelled from Liberia to the US on 19 September to join his girlfriend, Louise Troh, the mother of his son, Karsiah. After falling ill a few days later, Duncan was initially sent home from hospital, despite telling a nurse he had recently travelled from west Africa. He was taken by ambulance to Texas Health Presbyterian on 28 September, where he was admitted and placed in isolation.

He was confirmed to have Ebola two days later.

It is believed Duncan contracted the disease while helping take his landlord’s 19-year-old daughter to an Ebola treatment ward in Monrovia. He did not declare that he had been in contact with Ebola when he completed a pre-flight questionnaire at Monrovia airport before travelling to the US.

The current outbreak of Ebola has killed more than 4,000 people in west Africa.

In his opening remarks on Sunday, judge Jenkins sought to calm public fears. He said: “I want to stress an important fact. You cannot contract Ebola other than from bodily fluids of a symptomatic Ebola victim. You cannot contract Ebola by walking by people in the street or from contacts who are not symptomatic. There is nothing about this case that changes that basic premise of science.

“And so it’s important that while this is obviously bad news, it is not news that should bring about panic. We have a strategy to monitor this and we will go to that strategy to keep the community safe.”

The hospital said it was “triple-checking our full compliance with updated CDC guidelines. We are also continuing to monitor all staff who had some relation to Mr. Duncan’s care even if they are not assumed to be at significant risk of infection”.